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Seminar

GT Graduate Student Seminar: "From Hyperbolic Crystals to Bundles: Moduli Spaces in Spectral Theory" October 11, 2022 (10:00 AM PDT - 11:00 AM PDT)
Parent Program:
Location: MSRI: Simons Auditorium, Online/Virtual
Speaker(s) Elliot Kienzle (University of California, Berkeley)
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GT Graduate Student Seminar: "From Hyperbolic Crystals To Bundles: Moduli Spaces In Spectral Theory"

Abstract/Media

To participate in this seminar, please register HERE.

Periodic crystals on the hyperbolic plane underlie an emerging bridge between condensed matter physics and algebraic geometry. Mathematically, hyperbolic crystals prompt us to study the spectrum of the hyperbolic laplacian plus a potential which is periodic under some hyperbolic lattice. Motivated by band theory, the space of functions splits into representations of the hyperbolic lattice, decomposing the spectrum into "bands" over the moduli space of such representations. Geometrically, these bands are the spectrum of the laplacian of a flat connection on the associated Riemann surface, graphed over the moduli space of such connections.  After introducing this, I will discuss my own work (joint with Steve Rayan) incorporating Higgs bundles into the story.  Higgs bundles enjoy a couple natural spectral-theoretic interpretations, first appearing from complex representations, and second encoding symmetries of the underlying hyperbolic lattice. Time permitting, I'll daydream about how Higgs bundles might weave hyperbolic crystals into a web of ideas across mathematics and physics.

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GT Graduate Student Seminar: "From Hyperbolic Crystals To Bundles: Moduli Spaces In Spectral Theory"